Problem 81 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph of a function having the given properties. $$ f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)<0 \text { on }(-\infty, \infty) $$

Step-by-Step Solution

Answer

The graph of the function should show a point at (2,4), a local maximum at x=2, and a downward curve (concave down) on both sides of the local maximum. This is due to the given properties \(f(2) = 4\), \(f'(2) = 0\), and \(f''(x) < 0\) for all \(x\). Keep in mind that this is only a sketch and not an exact representation of the function.

Step 1: Plot the given point

Graph the point (2,4) since f(2) = 4.

Step 2: Determine the type of critical point

Since f'(2) = 0 and f''(x) < 0 for all x, we know that the function has a local maximum at x=2. Because the function is concave down everywhere, it will curve away from this maximum point.

Step 3: Sketch the graph using the given properties

Begin by drawing the local maximum at the point (2,4). Since the second derivative is negative everywhere, draw a downward curve (concave down) on both sides of the maximum. The sketch of the function should show: 1. A point at (2,4). 2. A local maximum at x=2. 3. A concave down curve across the whole graph. Remember, this is only a sketch and not an exact representation of the function, but it should visually convey the given properties.

Key Concepts

Critical Points

Critical points are where the function's derivative equals zero or is undefined. In simple terms, these points are where the function might stop increasing and begin to decrease, or vice-versa. Identifying these points is essential in understanding the overall behavior of a function.

In our exercise, we have a critical point at \( x = 2 \) because \( f'(2) = 0 \). This indicates a flat spot on the graph, where the tangent to the curve can be horizontal. A critical point does not always mean a maximum or minimum; it simply suggests potential changes in direction. Understanding the behavior around a critical point involves further analysis, like checking the second derivative, which we'll discuss in the next section.

Concavity

Concavity tells us how the graph of a function curves. Is it bending upwards or downwards? This is determined by the second derivative, \( f''(x) \).

- If \( f''(x) > 0 \), the graph is concave up, similar to a cup that can hold water.- If \( f''(x) < 0 \), the graph is concave down, like an upside-down cup.For our function, \( f''(x) < 0 \) over the entire range \((-\infty, \infty)\). This means the graph consistently curves downward. In other words, it is always concave down, never bending upwards in any region. This trait tells us the graph will always resemble an inverse arch or a frown, especially noticeable around any maximum points.

Graph Sketching

Graph sketching is an artistic yet analytical way to visualize how a function behaves. It combines all the analysis from critical points and concavity to draw a rough diagram of the function's graph.

Here are some general steps you might take:

• Identify and plot key points, like those where the function reaches maximums or minimums.
• Determine the overall shape of the graph using concavity information.
• Check where the graph crosses the axis if necessary, though not always required as in this example.

In the specific exercise, you start by plotting the given point at \((2,4)\), a local maximum because \( f'(2) = 0 \) and \( f''(x) < 0 \). You then sketch the curve with a downward concavity from this point. The graph bends away from this peak in both directions, with no upward bend anywhere.

Graph sketching helps in translating mathematical properties into a visual format, making it easier to understand and analyze functions.